Elliptic Curves over finite Fields of primepowers

[Shalec]

Hi,

I would like to set up an elliptic curve over a Field F_{p^n} for any prime p. Is there a way to do so? My suggestion is to setup an verifying algorithm for "Elliptic Curves" by Washington and the Theorem of Hasse. Therefor I would like to calc the Order of any elliptic curve over finite fields.

I just used those:

http://www.sagemath.org/de/html/tutorial/tour_advanced.html

Its not possible to use "GF(5^2)" and creating an elliptic curve over GF(5^2). Could someone tell me, how to realize this?

Thank you, 

Shalec

[John Cremona]

It *is* possible, and reading the documentation will show you many
similar examples:


sage: F.<a> = GF(5^2)
sage: E = EllipticCurve(F,[0,1,a,a,1])
sage: E.cardinality()
20
sage: E.abelian_group()
Additive abelian group isomorphic to Z/2 + Z/10 embedded in Abelian
group of points on Elliptic Curve defined by y^2 + a*y = x^3 + x^2 +
a*x + 1 over Finite Field in a of size 5^2
sage: E.cardinality(extension_degree=100)
62230152778611417071440640537801242405902521687211671331011166147896987511823444765142947526679666797869844792692444767708438246773316672000

John Cremona

>
> Thank you,
> Shalec
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